Suppose U is a subspace of V such that V/U is finite dimensional. Can we say that V is finite dimensional?

Question

Suppose U is a subspace of V such that V/U is finite dimensional. V/U is the quotient sapce, namely the set of all affine subsets of V parallel to U.

I think we cannot show that V is finite dimensional, but I am confused without any idea.

Answer 1

Here is a counterexample: let $V=\mathbb{k}[x,y]$, which is an infinite dimensional $\mathbb{k}$-vector space with a basis $\{ x^iy^j \mid i,j \ge 0\}$. Let $U$ be a subspace of $V$ generated by monomials with degree $\ge2$. Then we have that $V/U$ has dimension $=3$, with a basis $1,x,y$.